A257538 -id:A257538 - cp's OEIS frontend

A064988 Multiplicative with a(p^e) = prime(p)^e.

1, 3, 5, 9, 11, 15, 17, 27, 25, 33, 31, 45, 41, 51, 55, 81, 59, 75, 67, 99, 85, 93, 83, 135, 121, 123, 125, 153, 109, 165, 127, 243, 155, 177, 187, 225, 157, 201, 205, 297, 179, 255, 191, 279, 275, 249, 211, 405, 289, 363, 295, 369, 241, 375, 341, 459, 335, 327

Offset: 1

Vladeta Jovovic, Oct 30 2001

			a(12) = a(2^2*3) = prime(2)^2 * prime(3) = 3^2*5 = 45, where prime(n) = A000040(n).

Alois P. Heinz, Table of n, a(n) for n = 1..20000
Index entries for sequences computed from indices in prime factorization (first 1000 terms from Harry J. Smith)

Cf. A000040, A003961, A003963 (a left inverse), A006450, A048767, A257538, A290641.

Cf. A076610 (terms sorted into ascending order).

a:= n-> mul(ithprime(i[1])^i[2], i=ifactors(n)[2]):
seq(a(n), n=1..70);  # Alois P. Heinz, Sep 06 2018

Table[If[n == 1, 1, Apply[Times, FactorInteger[n] /. {p_, e_} /; p > 1 :> Prime[p]^e]], {n, 58}] (* Michael De Vlieger, Aug 22 2017 *)

{ for (n=1, 1000, f=factor(n)~; a=1; for (i=1, length(f), a*=prime(f[1, i])^f[2, i]); write("b064988.txt", n, " ", a) ) } \\ Harry J. Smith, Oct 02 2009

a(n) = {my(f = factor(n)); for (k=1, #f~, f[k, 1] = prime(f[k, 1]);); factorback(f);} \\ Michel Marcus, Aug 08 2017

from sympy import factorint, prime
from operator import mul
def a(n): return 1 if n==1 else reduce(mul, [prime(p)**e for p, e in factorint(n).items()])
print([a(n) for n in range(1, 101)]) # Indranil Ghosh, Aug 08 2017

(define (A064988 n) (if (= 1 n) n (* (A000040 (A020639 n)) (A064988 (A032742 n))))) ;; Antti Karttunen, Aug 08 2017

From Antti Karttunen, Aug 08 & 22 2017: (Start)
For n = p_{i1} * p_{i2} * ... * p_{ik}, where the indices i1, i2, ..., ik of primes p are not necessarily distinct, a(n) = A006450(i1) * A006450(i2) * ... * A006450(ik).
a(n) = A003961(A290641(n)).
A046523(a(n)) = A046523(n). [Preserves the prime signature of n].
A003963(a(n)) = n.
(End)

A295665 Fully multiplicative with a(prime(m)) = prime(k) when m = prime(k), and a(prime(m)) = 1 when m is not a prime.

Original entry on oeis.org

1, 1, 2, 1, 3, 2, 1, 1, 4, 3, 5, 2, 1, 1, 6, 1, 7, 4, 1, 3, 2, 5, 1, 2, 9, 1, 8, 1, 1, 6, 11, 1, 10, 7, 3, 4, 1, 1, 2, 3, 13, 2, 1, 5, 12, 1, 1, 2, 1, 9, 14, 1, 1, 8, 15, 1, 2, 1, 17, 6, 1, 11, 4, 1, 3, 10, 19, 7, 2, 3, 1, 4, 1, 1, 18, 1, 5, 2, 1, 3, 16, 13, 23, 2, 21, 1, 2, 5, 1, 12, 1, 1, 22, 1, 3, 2, 1, 1, 20, 9, 1, 14, 1, 1, 6

Offset: 1

Antti Karttunen, Nov 26 2017

The number of applications to reach 1 is A322027(n). Positions of first appearances are A076610. - Gus Wiseman, Jan 17 2020

			For n = 360 = 2^3 * 3^2 * 5 = prime(1)^3 * prime(2)^2 * prime(3), 1 is not a prime, but 2 and 3 are, thus a(360) = 2^2 * 3 = 12.

Antti Karttunen, Table of n, a(n) for n = 1..10000
Index entries for sequences computed from indices in prime factorization

Cf. A000040, A000720, A010051, A295666.
Cf. also A003963, A257538.
Positions of 1's are A320628.
Positions of terms > 1 are A331386.
Primes of prime index are A006450.
Primes of nonprime index are A007821.
Products of primes of prime index are A076610.
Products of primes of nonprime index are A320628.
The number of prime prime indices is A257994.
The number of nonprime prime indices is A330944.
Numbers whose prime indices are not all prime are A330945.
Cf. A001222, A007097, A018252, A056239, A112798, A320633, A322027, A330947, A330948.

Table[Times@@Cases[FactorInteger[n],{p_?(PrimeQ[PrimePi[#]]&),k_}:>PrimePi[p]^k],{n,40}] (* Gus Wiseman, Jan 17 2020 *)

(definec (A295665 n) (if (= 1 n) 1 (let ((k (A055396 n))) (* (if (zero? (A010051 k)) 1 k) (A295665 (A032742 n))))))

Multiplicative with a(p^e) = A000720(p)^(e*A010051(A000720(p))).
a(1) = 1; for n > 1, if A055396(n) is a prime, then a(n) = A055396(n) * a(A032742(n)), otherwise a(n) = a(A032742(n)).
Other identities. For all n >= 1:
a(A006450(n)) = A000040(n).
a(A007097(n)) = A007097(n-1).
a(A294876(n)) = A295666(n).

cp's OEIS Frontend

A064988 Multiplicative with a(p^e) = prime(p)^e.

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